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\title{微分方程模型习题 }

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\item  设室温为20度。将一根冰棍放在碗里，冰棍的初始温度为零下10度。
求冰棍的体积与时间的函数关系式。什么时候冰棍不见了？

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\item  例子8.3. 使用sympy模块的dsolve函数，求解微分方程的符号解， 
\begin{eqnarray}
y'=-2y+2x^2+2x,\,\, y(0)=1. 
\end{eqnarray}

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\item  例子8.8. 使用scipy.integrate模块的odeint函数，求解微分方程的数值解，
\begin{eqnarray}
5(1-x)y'' = \sqrt{1+(y')^2},\,\, y(0)=0,\,\, y'(0)=0.
\end{eqnarray}
首先将该二阶常微分方程写成一阶线性微分方程组的形式。


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\item  选取洛伦兹方程的参数 $\sigma,\rho,\beta$ 的不同取值，以及初始值的不同初值，使用odeint函数求解，并画出轨线图。
\begin{eqnarray}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& \sigma(y-x), \\ 
\frac{dy}{dt} &=& \rho x - y -xz, \\ 
\frac{dz}{dt} &=& xy - \beta z, \\ 
\end{array}\right. 
\end{eqnarray}



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\item  使用理论计算和程序计算两种方法，求解下述常微分方程的初值问题，画出解函数的图像。
\begin{eqnarray}
\frac{dx}{dt} = 2x, \,\, x(0)=10.  
\end{eqnarray}

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\item  使用理论计算和程序计算两种方法，求解下述常微分方程的初值问题，画出解函数的图像。
\begin{eqnarray}
\frac{dx}{dt} = 2x(20-x), \,\, x(0)=10.  
\end{eqnarray}

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\item  使用理论计算和程序计算两种方法，求解下述常微分方程组，画出相图。
\begin{eqnarray}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& 2x + 3y, \\
\frac{dy}{dt} &=& 2x - 3y. 
\end{array}\right.
%\label{eq-1}
\end{eqnarray}

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\item  使用理论计算和程序计算两种方法，求解下述常微分方程组，画出相图。
\begin{eqnarray}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& 3x, \\
\frac{dy}{dt} &=& 2x + y. 
\end{array}\right.
%\label{eq-1}
\end{eqnarray}

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\item  使用理论计算和程序计算两种方法，求解二阶线性常微分方程的初值问题，，画出解函数的图像。
\begin{eqnarray}
y''+y'-2y=2x, \,\, y(0)=0, y'(0)=1. 
\end{eqnarray}

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\item  查阅有关 Lotka-Volterra 模型的近十年的研究文献。选3-5篇按格式排版，并写一段文献简述。

\begin{table}[ht!]\centering
\caption{参考文献条目格式}\vspace{0.2cm}
\begin{tabular}{|p{2cm}|p{12cm}|}\hline 
文献类别&	文献条目格式 \\ \hline 
学术著作类&	[序号]作者.书名[M].版本（初版不写）.翻译者.出版社,出版年份.  \\ \hline 
学术期刊类&	[序号]作者.篇名[J].刊名,出版年份(期号):起止页码.  \\ \hline 
论文集类&	[序号]作者.篇名[A].论文集名[C],出版者,出版年份:起止页码．  \\ \hline 
科技报告类&	[序号]报告者.报告题目[R].报告地:报告会主办单位,报告年份.  \\ \hline 
学位论文类&	[序号]作者.论文名[D].博士（或硕士）学位论文.授予单位,授予年份． \\ \hline 
专利文献类&	[序号]申请者.专利名[P].国别.专利号,授权日期.  \\ \hline 
技术标准类&	[序号]发布单位.技术标准代号.技术标准名称[S].出版者,出版日期． \\ \hline 
报纸文献类&	[序号]作者.篇名[N].报纸名.出版日期（版面次序）．  \\ \hline 
电子文献类&	[序号]著者.文献题名.电子文献类型标示.文献网址或出处,发表日期． \\  
&  \hspace{2.7em} 电子参考文献建议标识：\\ 
&  \hspace{2.7em} [DB/OL] ——网上数据库 \\
&  \hspace{2.7em} [J/OL] ——网上期刊 \\ 
&  \hspace{2.7em} [EB/OL] ——网上电子公告  \\ \hline 
\end{tabular}
\end{table}

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文献简述：
在丁同仁的教材里介绍了捕食者被捕食者模型，给出了简单形式的微分方程模型的理论求解方法\cite{dingtongren}。 


\begin{thebibliography}{99}

\bibitem{dingtongren} 丁同仁,李承治. \emph{常微分方程教程} [M]. 第3版. 高等教育出版社, 2022年. 
\bibitem{sishoukui-2} 司守奎,孙玺菁. \emph{Python数学建模算法与应用} [M]. 国防工业出版社, 2022年. 
\bibitem{lorenz} Lorenz, E. N. \emph{Deterministic Nonperiodic Flow} [J]. Journal of the Atmospheric Sciences, 1963(20): 130-141.
\bibitem{wolfram} Wolfram MathWorld. \emph{Lorenz Attractor} [EB/OL].\\ \url{https://mathworld.wolfram.com/LorenzAttractor.html}. 

\end{thebibliography}


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\item  写出单摆的角度与时间的函数 $\theta(t)$ 所满足的微分方程。
\begin{enumerate}
\item  不考虑空气阻力。
\item  考虑空气阻力。
\end{enumerate}

\begin{center}
\includegraphics [height=4cm, width=5cm]{pendulum-2.png}
\end{center}

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\item  ** 设有一根粗细和密度都均匀的木杆，一端固定，从水平状态自由摆动。
设木杆的质量为 $m$, 长度为 $L$. 求木杆摆到竖直状态所需要的时间。

\begin{center}
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\end{center}

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\item  导出悬链线满足的微分方程，并写出边界条件。

\begin{center}
\includegraphics [height=5cm, width=9cm]{catenary.png}
\end{center}

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\item  用费马原理（光线总是沿最短时间的路径前进）证明光线的折射原理：光线在两种介质中的路径发生偏折，入射角和出射角的正弦的比例，等于光线在两种介质中的速度的比例，即 $\frac{\sin\alpha_1}{\sin\alpha_2} = \frac{v_1}{v_2}. $

\begin{center}
 \includegraphics[height=5cm, width=8cm]{fermat-refraction-snell-law.png}
\end{center}

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